I believe the answer to your question is that both Druids and Shamen rely on alternate planes of existence for their source of power. Druids use the sans-sentient blueprint of Azeroth, the Emerald Dream, to manipulate themselves and life around them. Shamen ask for aid from the elemental spirits bound in the Elemental Plane, and can also draw on animal spirits and their ancestors.
A Druid cannot cast a ball of fire, and a Shaman cannot feel the pain of a tree. Deepholm, the Earthen Plane of Therazane does not boast a great deal of living creatures. She has her mushrooms and bog giants, but beyond that everything is made of gems and stones. The living, breathing force of nature is the heart of the Emerald Dream and the realm of the Druid. The elements around us that are the fundamental ingredients for nature are the realm of the Shaman.
As The Dream is Azeroth proper void of sentient life, and the Elemental Plane is a separate dimension with no direct connection to The Dream I feel like this might answer your question about Fire Druids. Druids can control The Dream and parts of Azeroth, but they can't commune with the Elemental Plane. Shaman can commune with the Elemental Plane and control facets of Azeroth, but they cannot connect to The Dream. A Fire Druid is a Druid that can connect to the Emerald Dream and wield the powers of the Plane of Fire; and this is only possible because of Ragnaros' merging of the plane of fire with the natural world which puts the Plane of Fire under the influence of the Emerald Dream. Get it?
As to the Druids' connection with the spirits, I'm not really sure how that goes down. I took it that the Emerald Dream is a mirrored reflection of the living forces of the real world, and anything residing within is a doppelganging mirror of a true self, rather than a deceased incorporeal form. Shamen on the other hand specifically deal with the deceased in the spirit world, and the spirits they call upon that allow them to speak with ancestors and summon ghost wolves have nothing to do with The Dream or Elemental Plane but rather the plane of the dead, which predates those Titan constructed realms.
But it gets messy no matter how you slice it because Druids still deal with tangible deceased spirits of nature, aka wisps, and Night Elves even become wisps when they die. Troll Druids even draw their power from Loa, deceased spirits revered by their people, and these Loa (I.E. the raptor Gonk) know the secrets of the Emerald Dream. My best guess for you is that the Titans somehow built a bridge between the spirit world and the Emerald Dream that allows the animal spirits and people especially in tune with nature to freely pass between the netherworld and The Dream after death. This is why Night Elves, unlike say Orcs or Humans, can become one with nature upon death as a nature spirit (wisp), and even be reborn as Ancients. This would also explain why the Trolls in Moonglade refer to wisps and Ancient Guardians as Loa.
As for their command of storms and hurricanes, I honestly have nothing for you. But if I absolutely had to guess I would tie it to Elune, whatever she is. Perhaps the Emerald Dream isn't just Azeroth, but the natural satellite, nearest star and solar system proper which all affect the nature and life of the planet. I would say that Druids call upon the gravitational fields of the Moon and Sun to induce small storms on Azeroth, which both forces can do, and that is how they shape the wind without directly connecting to the spirits of air and The Skywall.
Please gods this. I don't know about Druids controlling the elements without the Elemental Planes merging into Azeroth and thus falling under the Emerald Dream, but I've spent the past two days swimming around as an orca in Vashj'ir and I would kill for a water themed expansion and some more water forms. After all, oceanic life is still an essential part of nature and there is no reason Druids should not commune with it more strongly. Druids of the Claw? Druids of the Talon? How about the Druids of the Flipper? That's what I'm talking about! Turning into a Feral shark, calling on dolphins like Aquaman. Mmmmm.